A finite simplicial complex $K$ is the union of subcomplexes $L$ and $M$. Describe the Mayer-Vietoris exact sequence that relates the homology groups of $K$ to those of $L$, $M$ and $L \cap M$. Define all the homomorphisms in the sequence, proving that they are well-defined (a proof of exactness is not required).

A surface $X$ is constructed by identifying together (by means of a homeomorphism) the boundaries of two Möbius strips $Y$ and $Z$. Assuming relevant triangulations exist, determine the homology groups of $X$.